Israeli tanks and thousands of troops today battled Hamas fighters as they pushed deep into the Gaza strip in a dramatic escalation of the conflict which has killed more than 30 Palestinians and one Israeli soldier since the ground invasion began.

Soldiers reached as far as the Mediterranean coast, cutting Gaza in half and seizing control of large areas of the overcrowded territory.

Israel's ground operation, which had been widely expected, began on Saturday night, the eighth of the conflict, and is the biggest assault on Gaza since Israel withdrew its Jewish settlers in 2005.

Television footage showed troops wearing night vision goggles, their faces painted in camouflage, marching in single file across the border.

Palestinian medical sources said more than 30 Palestinians had been killed in the offensive since midnight – three Hamas fighters and the rest civilians. The Israeli soldier was killed by mortar fire early this morning in northern Gaza, the first Israeli fatality in the ground offensive, the army said.

Thick clouds of smoke hung over Gaza as the Israeli attack targeted the northern towns of Beit Hanoun, Beit Lahiya and Jabaliya, all of which have been frequent scenes of Israeli incursions over recent years.

Israeli troops were also seen near the town of Rafah, in the southern Gaza Strip along the Egyptian border.

The streets of Gaza City were deserted and the city was surrounded by Israeli forces to the north, east and south.

Naval ships in the Mediterranean continued to fire shells into Gaza, along with artillery rounds from the east and repeated air strikes across the length of the territory.

Tens of thousands of Israeli reservists have been called up – a sign that the operation could yet deepen.

Said Judeh, of the Kamal Adwan hospital in Beit Lahiya, said eight people had been killed in an Israeli artillery strike. He said they were among a group of people trying to leave their homes to shelter in a nearby school.

Beit Lahiya is the scene of some of the heaviest fighting, and it was reported today that five members of the same family had been killed in two separate artillery strikes on their home there.

At least 30 Israeli troops were injured after the invasion began, with two, a soldier and an officer, seriously hurt.

The Hamas al-Aqsa television channel today reported that Hamas fighters had captured two Israeli soldiers, but the Israeli army said it had no knowledge of any such incident and that previous Hamas reports of Israeli casualties had proved inaccurate.

The Israeli defence minister, Ehud Barak, said the Israelis were "peace seekers". "We are not war hungry, but we shall not, I repeat shall not, allow a situation where our towns, villages and civilians are constantly targeted by Hamas," he said. "It will not be easy or short, but we are determined."

Brigadier General Avi Benayahu, an Israeli military spokesman, said the military's goals in launching what it called "phase two" of its campaign in Gaza were "to deal a heavy blow to the Hamas terror organisation, to strengthen Israel's deterrence and to create a better security situation for those living around the Gaza Strip that will be maintained for the long term".

The decision to launch a ground offensive came after a late-night meeting between the prime minister, Ehud Olmert, Barak and the foreign minister, Tzipi Livni, which lasted until 4am on Saturday.

A security cabinet meeting then approved the operation, although two ministers reportedly abstained.

The development followed another day of intense Israeli bombing in Gaza on Saturday. Among the targets destroyed was the American International School in northern Gaza, a private school which has been attacked by Palestinian militants in the past.

Another Israeli air strike destroyed a mosque in Beit Lahiya, killing around a dozen Palestinians.

The death toll in Gaza climbed to nearly 500, with more than 2,300 wounded, according to Palestinian medical officials.

Four Israelis have died in rocket and mortar attacks from Gaza since Saturday, with around 60 wounded. Hamas launched further rocket attacks on Israel today.

Hamas leaders warned of heavy fighting to come. "You entered like rats. Your entry to Gaza won't be easy. Gaza will be a graveyard for you, God willing," Ismail Radwan, a Hamas spokesman said in a statement broadcast on Hamas television.

In Damascus, the Hamas leader in exile, Khaled Meshal, said on Friday that Israeli soldiers faced a "black fate" in Gaza.

He said Hamas's conditions for a ceasefire were that Israeli attacks stopped, that Israel lifted its long economic blockade of Gaza and that the Rafah border crossing into Egypt be opened for people and goods.

Israel is believed to want tougher conditions, including a halt to Palestinian rocket fire, an end to smuggling, continued control of the crossings into Gaza, and is thought likely to push for international monitoring of any ceasefire deal.

Diplomatic efforts to end the conflict were appearing to falter. Late on Saturday night, the US blocked the approval of a UN security council statement calling for an immediate ceasefire.

Alejandro Wolff, the US deputy ambassador, said Hamas would not agree to halt the violence and that a new statement "would not be adhered to and would have no underpinning for success, would not do credit to the council".

The UN secretary general, Ban Ki-Moon, spoke of his "extreme concerns and disappointment" about the invasion, and France also condemned Israel's ground attack.

However, Jiri Potuznik, a spokesman for the EU presidency, which is currently held by the Czech Republic, issued a statement that was strikingly supportive of Israel, saying: "At the moment, from the perspective of the last days, we understand this step as a defensive, not offensive, action."